Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \frac{\frac{1}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \frac{\frac{1}{3}}{\sqrt{x}}
double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x))));
}
↓
double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - (y * ((1.0 / 3.0) / sqrt(x))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\sqrt{\color{blue}{1 \cdot x}}}\]

  6. Applied sqrt-prod0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\color{blue}{\sqrt{1} \cdot \sqrt{x}}}\]

  7. Applied div-inv0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\color{blue}{y \cdot \frac{1}{3}}}{\sqrt{1} \cdot \sqrt{x}}\]

  8. Applied times-frac0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{y}{\sqrt{1}} \cdot \frac{\frac{1}{3}}{\sqrt{x}}}\]

  9. Simplified0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{y} \cdot \frac{\frac{1}{3}}{\sqrt{x}}\]

  10. Final simplification0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \frac{\frac{1}{3}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))